Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

An activity making various patterns with 2 x 1 rectangular tiles.

This activity investigates how you might make squares and pentominoes from Polydron.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Can you use the information to find out which cards I have used?

How many models can you find which obey these rules?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

What is the best way to shunt these carriages so that each train can continue its journey?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Can you find all the different ways of lining up these Cuisenaire rods?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

What could the half time scores have been in these Olympic hockey matches?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

How many trains can you make which are the same length as Matt's, using rods that are identical?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Find out what a "fault-free" rectangle is and try to make some of your own.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the different ways you could split up these rooms so that you have double the number.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?